
A Hilbert subset of \(\mathbb {A}^n_K\) , as defined in [1]} and [2]}, is the intersection of finitely many dense open subsets of \(\mathbb {A}^n_K\)  and finitely many sets \(U_{f,K}\) .
In particular, the complement of a Hilbert subset in \(\mathbb {A}^n_K(K)\)  is a thin subset.
It is shown in [1]} and [2]} that every Hilbert subset of \(\mathbb {A}^n_K\)  contains an \(R\) -integral point,
i.e., the image of \(\mathbb {A}^n_R(R) \rightarrow \mathbb {A}^n_K(K)\)  is not thin.
Therefore, the arithmetic scheme \(\mathbb {A}^n_R\)  has the Hilbert property over \(R\) . (This also follows for all \(n\ge 2\)  from the Hilbert property of \(\mathbb {A}^1_R\)  over \(R\)  and Theorem REF .) Of course then \(\mathbb {P}_R^n\)  also has the Hilbert property over \(R\) , as it contains \(\mathbb {A}^n_R\)  as a dense open subscheme.
